Run-to-run (R2R) control has been widely used in semiconductor fabrication facilities for controlling various processes such as overlay, photo-resist, etching, chemical-mechanical-planarization (CMP) process, or diffusion. R2R control refers to a process control in which a recipe with respect to a particular fabrication process is modified between runs to minimize the impacts of process drift. Inputs and outputs of each process run are taken into account by an R2R controller at the end of each run. With this feedback, the R2R controller is able to obtain an optimal recipe for the next run and thereby enhance the performance.
An effective way to implement an R2R control system is to use a process model. Most process models for R2R control are expressed as a linear relation between inputs and outputs. An exemplary expression of an R2R control model may be as follows:γk=buk−1+dk−1 
where yk represents an output, uk−1 represents an input, b represents a model slope, and dk−1 represents an offset (i.e., y-intercept).
A fabrication process is usually associated with different contexts (e.g., different fabrication tools, reticles, products, fabrication layers, etc.) that represent components of the fabrication process. Accordingly, the offset dk−1 used in the above expression depends upon the specific contexts associated with the fabrication process and is obtained through experimentation. However, it is complicated and expensive to determine each offset value through experimentation. Thus, most R2R control systems estimate offsets based on pilot runs. A pilot run refers to a test process performed on a wafer, through which an initial offset estimation can be obtained.
Currently, in R2R control systems, a pilot run is preformed for each thread to estimate an offset of the thread. A thread refers to a specific run of the process in relation to a combination of contexts. If at least one context in a thread is changed, a new pilot run may be performed to estimate an offset for this thread. A pilot run may also need to be repeated periodically since the offset may drift during the fabrication process. An additional pilot run may be further needed after a maintenance check.
However, performing a pilot run for every thread in the fabrication process is expensive and time-consuming, especially for a “highly-mixed” semiconductor fabrication, in which numerous different products are manufactured. A typical “highly-mixed” fabrication manufactures approximately 150 different products. If each thread corresponds to a unique combination of a product and a tool, and there are 150 different products and 2 different tools, at least 150×2=300 pilot runs will be required. If a thread corresponds to a unique combination of a product, a tool and a layer, and there are 150 different products, 2 different tools and 2 different layers, then the minimum number of pilot runs will be 150×2×2=600. The same is true for “low-runner” products, for which only a few lots are processed, because a separate pilot run will still be required for each thread in the fabrication process.